In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?
Since they only ask for one length, you can assume that there's only one possible triangle.  Then, quickly note that $39 = 3\cdot 13$, and that 5 - 12 - 13 is a Pythagorean triple.  Therefore, the shorter leg has length $\boxed{15}$.